A photograph of the British countryside has been placed among the oil paintings in the National Gallery – causing people idling past to do double-takes. You can see them wondering: “Is that a painting? It’s so smooth, so shiny, so flat.” The picture shows bright fields and skies, seen through a dark thicket: the sensation of looking out from a hidden nook makes it an introspective, hesitant work. Perhaps that’s why it hangs so well alongside a great landscape painting by the quiet and contemplative master of the genre, John Constable.

Richard Billingham’s shot has invaded the holy sanctum of high art that is the National’s permanent collection as part of Seduced By Art, an exhibition of past and present photography. The gallery is a temple to oil on canvas. What happens when you allow photographs among the daubs?

Billingham strikes up a sombre, sensitive conversation with Constable’s The Cornfield: the result is a comparison of the English countryside in the early 1800s and early 2000s, proving, as the show claims, that photography can have a meaningful relationship with great painting. That’s just as well, because the two other pairings in the main galleries are disastrous. Seeing Richard Learoyd’s photograph Jasmijn in Mary Quant next to Ingres’s 19th-century beauty Madame Moitessier does nothing for either. As for a Craigie Horsfield photographic nude, shown between two sensual paintings by Degas, it’s an elephant among elegance.

Yet Degas, as it happens, was fascinated by photography. The great 19th-century painter of modern life took photographs and brooded on the relationship between the brush and the camera. Elsewhere in the National hangs his portrait of Princess Pauline de Metternich. Based on a photograph, the work gives the princess the slightly cadaverous look of some Victorian snaps, almost as if he’d copied a 19th-century deathbed photo. It’s a shame the show did not place some of Degas’s own photographs among his paintings, instead of Horsfield’s grimly ponderous black-and-white nude; this life-size work was surely inspired by After the Bath, Woman Drying Herself, the Degas work it partners (and thus, in turn, inspired by Degas’s own source, a drawing by Michelangelo). But arty quotation does not make art more powerful.

Is photography art? Clearly, some photography is, along with millions of camera-made images – from passport pictures to surveillance stills to my own snapshots – that are not. The trouble with Seduced By Art is that it has selected photographs that clearly aspire to be Art with a capital A. But why not put a passport photograph next to Giovanni Bellini’s Renaissance portrait of Doge Lorenzo Loredan? It is, after all, an exact depiction of someone’s facial features. Or what about a holiday snap next to something by the great 17th-century landscapist Claude Lorrain? That might say more, suggest more, matter more.